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A set of generators (g_1,...,g_n) is a set of group elements such that possibly repeated application of the generators on themselves and each other is capable of producing ...

A group homomorphism is a map f:G->H between two groups such that the group operation is preserved:f(g_1g_2)=f(g_1)f(g_2) for all g_1,g_2 in G, where the product on the ...

An element of order 2 in a group (i.e., an element A of a group such that A^2=I, where I is the identity element).

The kernel of a group homomorphism f:G-->G^' is the set of all elements of G which are mapped to the identity element of G^'. The kernel is a normal subgroup of G, and always ...

In celestial mechanics, the fixed path a planet traces as it moves around the sun is called an orbit. When a group G acts on a set X (this process is called a group action), ...

The number of elements in a group G, denoted |G|. If the order of a group is a finite number, the group is said to be a finite group. The order of an element g of a finite ...

Let Gamma be a representation for a group of group order h, then sum_(R)Gamma_i(R)_(mn)Gamma_j(R)_(m^'n^')^*=h/(sqrt(l_il_j))delta_(ij)delta_(mm^')delta_(nn^'). The proof is ...

A presentation of a group is a description of a set I and a subset R of the free group F(I) generated by I, written <(x_i)_(i in I)|(r)_(r in R)>, where r=1 (the identity ...

For any prime number p and any positive integer n, the p^n-rank r_(p^n)(G) of a finitely generated Abelian group G is the number of copies of the cyclic group Z_(p^n) ...

A representation of a group G is a group action of G on a vector space V by invertible linear maps. For example, the group of two elements Z_2={0,1} has a representation phi ...